In this paper, we classify del Pezzo foliations of rank at least 3 on projective manifolds and with log canonical singularities in the sense of McQuillan.
A smooth foliation is Riemannian when its leaves are locally equidistant. The closures of the leaves of a Riemannian foliation on a simply-connected manifold, or more generally of a Killing ...foliation, are described by flows of transverse Killing vector fields. This offers significant technical advantages in the study of this class of foliations, which nonetheless includes other important classes, such as those given by the orbits of isometric Lie group actions. Aiming at a broad audience, in this survey we introduce Killing foliations from the very basics, starting with a brief revision of the main objects appearing in this theory, such as pseudogroups, sheaves, holonomy and basic cohomology. We then review Molino’s structural theory for Riemannian foliations and present its transverse counterpart in the theory of complete pseudogroups of isometries, emphasizing the connections between these topics. We also survey some classical results and recent developments in the theory of Killing foliations. Finally, we review some topics in the theory of singular Riemannian foliations, including the recent proof of Molino’s conjecture, and discuss singular Killing foliations.
We consider smooth codimension q foliations on n-dimensional manifolds where 0<q<n. We use Ehresmann connections as a technical tool to introduce the notion of sensitivity to initial conditions for ...foliations. We extend Devaney's definition of chaos for cascades to foliations with Ehresmann connection. Our main result states that sensitivity to initial conditions of a foliation with Ehresmann connection follows from topological transitivity and density of minimal sets of the foliation. Compactness both minimal sets and the ambient manifold is not assumed. The results are applied to complete Cartan foliations.
We investigate the notion of the p-divisor for foliations on a smooth algebraic surface defined over a field of positive characteristic p and we study some of its properties. We present a structure ...theorem for the p-divisor of foliations in the projective plane and the Hirzebruch surfaces where we show that, under certain conditions, such p-divisors are reduced.
We show that a singular Riemannian foliation of codimension two on a compact simply-connected Riemannian
(
n
+
2
)
-manifold, with regular leaves homeomorphic to the
n
-torus, is given by a smooth ...effective
n
-torus action. This solves in the negative for the codimension 2 case a question about the existence of foliations by exotic tori on simply-connected manifolds.
The aim of this paper is to study codimension one foliations on rational homogeneous spaces, with a focus on the moduli space of foliations of low degree on Grassmannians and cominuscule spaces. ...Using equivariant techniques, we show that codimension one degree zero foliations on (ordinary, orthogonal, symplectic) Grassmannians of lines, some spinor varieties, some Lagrangian Grassmannians, the Cayley plane (an E6-variety) and the Freudenthal variety (an E7-variety) are identified with restrictions of foliations on the ambient projective space. We also provide some evidence that such results can be extended beyond these cases.
We show that a compact manifold that admits a Killing foliation with positive transverse curvature fibers over finite quotients of spheres or weighted complex projective spaces provided that the ...singular foliation defined by the closures of the leaves has maximal dimension. This result is obtained by deforming the foliation into a closed one while maintaining transverse geometric properties, which allows us to apply results from the Riemannian geometry of orbifolds to the space of leaves. We also show that the basic Euler characteristic is preserved by such deformations. Using this fact, we prove that a Riemannian foliation of a compact manifold with finite fundamental group and nonvanishing Euler characteristic is closed. As another application, we obtain that, for a positively curved Killing foliation of a compact manifold, if the structural algebra has sufficiently large dimension, then the basic Euler characteristic is positive.
The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many ...important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students.
Let ι:F0→F1 be a suitably oriented inclusion of foliations over a manifold M, then we extend the construction of the lower shriek maps given by Hilsum and Skandalis to adiabatic deformation groupoid ...C*-algebras: we construct an asymptotic morphism (ιad0,1))!∈En(C⁎(Gad0,1)),C⁎(Gad0,1))), where G and H are the monodromy groupoids associated with F0 and F1 respectively. Furthermore, we prove an interior Kasparov product formula for foliated ϱ-classes associated with longitudinal metrics of positive scalar curvature in the case of Riemannian foliated bundles.
PDF
